Table of Contents
Advances in Numerical Analysis
Volume 2014 (2014), Article ID 109525, 10 pages
http://dx.doi.org/10.1155/2014/109525
Research Article

The Perturbation Bound for the Spectral Radius of a Nonnegative Tensor

1School of Mathematical Sciences, South China Normal University, Guangzhou, China
2Department of Mathematics, Hong Kong Baptist University, Hong Kong

Received 29 March 2014; Revised 29 August 2014; Accepted 9 September 2014; Published 28 September 2014

Academic Editor: Nils Henrik Risebro

Copyright © 2014 Wen Li and Michael K. Ng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the perturbation bound for the spectral radius of an mth-order n-dimensional nonnegative tensor . The main contribution of this paper is to show that when is perturbed to a nonnegative tensor by , the absolute difference between the spectral radii of and is bounded by the largest magnitude of the ratio of the ith component of and the ith component , where is an eigenvector associated with the largest eigenvalue of in magnitude and its entries are positive. We further derive the bound in terms of the entries of only when is not known in advance. Based on the perturbation analysis, we make use of the NQZ algorithm to estimate the spectral radius of a nonnegative tensor in general. On the other hand, we study the backward error matrix and obtain its smallest error bound for its perturbed largest eigenvalue and associated eigenvector of an irreducible nonnegative tensor. Based on the backward error analysis, we can estimate the stability of computation of the largest eigenvalue of an irreducible nonnegative tensor by the NQZ algorithm. Numerical examples are presented to illustrate the theoretical results of our perturbation analysis.